Question
Answer
$$(1+x+2x^2)^3=8x^6+12x^5+18x^4+13x^3+9x^2+3x+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+x+2x^2)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}8x^6+12x^5+18x^4+13x^3+9x^2+3x+1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1+x+2x^2}\right) $ by each term in $ \left( 1+x+2x^2\right) $. $$ \left( \color{blue}{1+x+2x^2}\right) \cdot \left( 1+x+2x^2\right) = 1+x+2x^2+x+x^2+2x^3+2x^2+2x^3+4x^4 $$ |
| ② | Combine like terms: $$ 1+ \color{blue}{x} + \color{red}{2x^2} + \color{blue}{x} + \color{green}{x^2} + \color{orange}{2x^3} + \color{green}{2x^2} + \color{orange}{2x^3} +4x^4 = 4x^4+ \color{orange}{4x^3} + \color{green}{5x^2} + \color{blue}{2x} +1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{4x^4+4x^3+5x^2+2x+1}\right) $ by each term in $ \left( 1+x+2x^2\right) $. $$ \left( \color{blue}{4x^4+4x^3+5x^2+2x+1}\right) \cdot \left( 1+x+2x^2\right) = \\ = 4x^4+4x^5+8x^6+4x^3+4x^4+8x^5+5x^2+5x^3+10x^4+2x+2x^2+4x^3+1+x+2x^2 $$ |
| ④ | Combine like terms: $$ \color{blue}{4x^4} + \color{red}{4x^5} +8x^6+ \color{green}{4x^3} + \color{orange}{4x^4} + \color{red}{8x^5} + \color{blue}{5x^2} + \color{red}{5x^3} + \color{orange}{10x^4} + \color{green}{2x} + \color{orange}{2x^2} + \color{red}{4x^3} +1+ \color{green}{x} + \color{orange}{2x^2} = \\ = 8x^6+ \color{red}{12x^5} + \color{orange}{18x^4} + \color{red}{13x^3} + \color{orange}{9x^2} + \color{green}{3x} +1 $$ |
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